Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively. Then, $\frak{t}$ can be considered as a Cartan subalgebra of $\frak{g}$. It is known that the kernel of exponential map $exp : \frak{t} \to$ $T$ is the lattice of all integral weights of $\frak{g}$, i.e. weihts $\lambda \in (it)^*$ such that $\lambda(H)\in 2\pi i\mathbb{Z},$ whenever $exp H= I$ for $H\in\frak{t}$.

I have the following questions:

1. What is the relation between the fundamental group $\pi_{1}(G)$ of $G$ with the integral lattice described above? I am trying to find any good references about this fact, but it seems difficult.

2. How we can use the fibration $T\to G$$\to G/T$ to compute $\pi_{1}(G/T)$? (answered)

3. What we can say about the second homotopy group $\pi_{2}(G)$? (answered)

4. Is it true, that if $G$ is semisimple, then $\pi_{1}(G)$ is finite? (answered)

Thank you!